$\renewcommand{\hole}{\phantom{x}}$
Have I mentioned that I “don’t like” simplicial sets? That I “really don’t like” Kan complexes? Neither squote is really true, of course, but simplicial sets and Kan complexes even more so, are not things you want ever to have to deal with by hand.

Benno $\mathrm{v^a_n d^e_n}$Berg

has noted that the adjunction
$$ \frac{\mathrm{Forget Degeneracy Maps} \dashv \mathrm{Weird Right Adjoint}}{ \mathrm{ssSet} \leftrightarrows \mathrm{sSet}} $$
allows transfer of the Kan model structure on $\mathrm{sSet}$ to $\mathrm{ssSet}$; and in particular, he identifies the semisimplicial sets that happen to be simplicial sets as the

*cofibrant* objects of this transfered model structure; and in particular, the semisimplicial sets that happen to be Kan simplical sets are the

*fibrant cofibrant* objects. For some unknown reason, this makes me happy!

For another, more perspicuous reason, it annoys the heck out of me, because the right adjoint is, well, the

*wrong functor* to simplicial sets from semisimplicial sets. In this model structure, only semisimplicial sets that can be made simplicial actually have “the right” cohomology, for one thing. Dropping the silly names, the obvious functor from simplicial to semisimplicial shall be called $i^*$; its right adjoint, the one van den Berg uses, shall be called $i_*$. The adjunction property specializes to
$$ Hom_{sss} (i^* [-,m],X) \simeq Hom_{ss} ( [-,m], i_* X) \simeq_\mathrm{Yoneda} (i_* X)_m $$
or, in other words, an element of $(i_* X)_m$ is an ordinary $m$-cell in $X$ together with

*a compatible choice of all its degenerations*. The cofibrant replacement of any

*finite* semisimplicial set is the empty simplicial set, for instance.

van den Berg also shows that lots of semisimplicial maps that I would have no objection to otherwise

*cannot* be cofibrations in

*any* useful model category structure on semisimplicial sets.

*Coloured* semisimplicial objects (just to have it written somewhere, and keep things interesting) are the functors on the walking augmented functor; this latter is the free $(\infty,2)$-category on a single object $*$, a morphism $ T : * \to *$ and a transformation $ \eta:T \to 1 $. The important diagrams in the category of morphisms thus generated start
$$ \xymatrix{ 1 & T \ar[d]|\eta & T^2 \ar[r]^{T\eta} \ar[d]|\eta & T \ar[d]|\eta & T^3 \ar[dd]|\eta \ar[rr]|{T^2\eta} \ar[dr]|{T\eta} & & T^2 \ar[dr]|{T\eta} \ar[dd]|\hole|(.6)\eta & & \cdots \\ & 1 & T \ar[r]_\eta & 1 & & T^2\ar[dd]|(.4)\eta \ar[rr]|(.4){T\eta} & & T\ar[dd]|\eta\\
& & & & T^2 \ar[dr]|\eta \ar[rr]|\hole|(.6){T\eta} & & T \ar[dr]|\eta \\
& & & & & T \ar[rr]|\eta & & 1 } $$ which should all look very familiar, as underlying the (co?)bar delooping construction, only we don’t say now that $T^3$ is $(T)^3$…
Ordinary semisimplicial sets are coloured simplicial sets that preserve the terminal object.

The Kan condition, which makes an $\mathrm{sSet}$

*fibrant*, might seem a little strange. What it “really” means, when you finish doing induction on skeleta, is that

*every finite contractible diagram* of simplices, in a Kan complex, can be embedded in

*a single* simplex with just the right number of vertices. That’s a lot of extra room to move around in!

(semi)Simplicial

*Sets* (read “h-Sets”, read “discrete space”) are inadequate for studying fibrations: there are no interesting (new) semisimplicial sets (fibered) over $\sph^2$, just because there are no interesting covers of $\sph^2$. There are plenty of weird fibrations over $\sph^2$, as any algebraic geometer can tell you, but one must consider at least (semi)simplicial $1$-types to get at them. One can alternatively view this as

another inadequacy to pretending that “everything” “is” a CW complex.